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Table of contents
I Prerequisites
1 Algebra
1.1 Linear algebra
1.2 Basic Galois theory
1.3 Basic algebraic geometry
2 Algorithms and complexity
2.1 Asymptotic complexity
2.2 Fundamental algorithms
II The transposition principle
3 Algebraic Complexity and duality
3.1 Arithmetic circuits
3.2 Multilinearity
3.3 Straight Line Programs
3.4 Automatic differentiation
4 Automatic transposition of code
4.1 Inferring linearity
4.2 transalpyne
III Fast arithmetics using univariate representations
5 Trace computations
5.1 Decomposition of a zero-dimensional ideal
5.2 Trace formulas
5.3 Sitckelberger’s theorem
5.4 Rational Univariate Representation
5.5 The univariate case
5.6 Shoup’s algorithm
5.7 From univariate to bivariate and back again
6 Artin-Schreier towers
6.1 Introduction
6.2 Preliminaries
6.3 A primitive tower
6.4 Level embedding
6.5 Frobenius and pseudotrace
6.6 Arbitrary towers
6.7 Experimental results
IV Applications to isogenies and cryptography
7 Elliptic curves and isogenies
7.1 Definitions
7.2 Curves over C
7.3 Curves over finite fields
7.4 Modular polynomials
8 Computing isogenies over finite fields
8.1 Overview
8.2 Vélu formulas
8.3 BMSS
8.4 Lercier-Sirvent
8.5 Couveignes’ algorithm
8.6 The algorithm C2-AS
8.7 The algorithm C2-AS-FI
8.8 The algorithm C2-AS-FI-MC
8.9 Isogenies of unknown degree
9 Experimental results
9.1 Implementation of Couveignes’ algorithm
9.2 Implementation of C2-UD
9.3 Implementation of Lercier-Sirvent
9.4 Benchmarks
A Categorical considerations
A.1 Categorical semantics of arithmetic circuits
A.2 Coevaluation
A.3 The tranposition theorem
A.4 From circuits to function-level programming
A.5 Self-transposing polynomial multiplication
B Linearity inference of Karatsuba multiplication
C Proof of Vélu’s formulas
Conclusion
List of symbols
Index
Bibliography
